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(EVn)   For all V e 93 and all A ^ 0 in R (resp. C), we have AV e SB.
(EVm)   For each V e $8, there exists W e SB such that W + W <= V.

Show that there exists a unique topology on E, compatible with the vector space
structure, for which 58 is a fundamental system of neighborhoods of 0.

2.    Let E be a vector space over R with a denumerable infinite basis (>), and let % be the
set of all balanced absorbing sets in E. Show that 5E does not satisfy axiom (EVm) of

Problem 1. (For each n > 0, let An be the set of points ]T tt et such that \tt\ < I In for


1 Iz-i^n* anc* let A be the union of the sets An . Show that there exists no set
M e % such that M + M c A.)

3.    Let E be the vector space #R(R) of continuous mappings of R into R. For each
continuous function m such that m(t) > 0 for all / e R, let Vm denote the set of functions
/e E such that |/(/)| g m(t) for all t e R.

(a)    Show that the Vm form a fundamental system of neighborhoods of 0 in a topology
^"0 on E which is compatible with the additive group structure of E, but not with the
vector space structure.

(b)    Let D be the vector subspace of E consisting of functions with compact support
(12.6). Show that the topology f induced by ^~0 on D is compatible with the vector
space structure of D, and is not metrizable.

4.    Let ()! be any family of topological vector spaces, and E = I"I E the product

vector space. Show that the product of the topologies of the E (1 2.5) is compatible with
the vector space structure of E. If each Ea is locally convex and its topology is defined
by a family of seminorms CpcuXieL,,, then the topology of E is locally convex and is
defined by the family of seminorms /?aA  pr (for all a e I and all A e L^), In particular,
every (finite or) denumerable product of Fre"chet spaces is a Fre"chet space.

5.    Let p be a seminorm on a (real or complex) vector space E. Show that the set N
of x e E such that p(x) === 0 is a vector subspace of E. The seminorm p defined by
( ) on E/N is a norm.

Let E be a Hausdorff locally convex space whose topology is defined by a family
(/?) of seminorms, and for each a let Na be the subspace of E consisting of those
x e E for which px(x) = 0. Show that E is isomorphic to a subspace of the product
of the normed spaces E/N (the norm on E/N being p*). In particular, every Fre"chet
space is isomorphic to a closed subspace of a denumerable product of normed spaces.

6.    A subset A of a vector space E over R is said to absorb a subset B of E if there exists
a > 0 such that AB c A whenever |A| :g a. If E is a topological vector space, a subset
B of E is said to be bounded (with respect to the topological vector space structure of E)
if every neighborhood of 0 absorbs B.

(a)    The closure of a bounded set is bounded. Every finite set is bounded. If A, B are
bounded, then so are A u B and A -f- B. If E is metrizable, every precompact set in
E is bounded.

(b)    Suppose that E is Hausdorff. Then a subset B of E is bounded if and only if,
for every sequence (xn) in B and every sequence (An) of real numbers tending to zero,
the sequence (Xnxn) tends to 0 in E.

(c)    Let (E) be a family of topological vector spaces and E ~ J"I E their product


(Problem 4). Show that a subset B of E is bounded if and only if pra(B) is bounded
in E for each a.each 1] into G is continuous. Deduce that this mapping can be extended to a non-